I'm reading the book "Fourier Analysis and Its Applications" and in the derivation of the Fourier transform he began with writing the Fourier series $$f(t)=\sum_{n=-\infty }^{\infty }c_{n}exp(in\frac{\pi}{P}t)$$
where $$c_{n}=\frac{1}{2P}\int_{-P}^{P}f(t)exp({-in\frac{\pi}{P}t)}dt$$
so we're projecting our function on the orthogonal basis in infinite dimensional function space
he then defined a function $$f (P, ω) = \int_{-P}^{P}f(t)e^{-iwt}dt$$ then he re wrote the Fourier series as $$f(t)=\frac{1}{2\pi}\sum_{n=-\infty }^{\infty }f (P, ω_{n})exp(iw_{n}t).\frac{\pi}{P},, w_{n}=\frac{n\pi}{P}$$
which is equal $$f(t)=\sum_{n=-\infty }^{\infty }\frac{\Delta w}{2\pi} \begin{pmatrix} \int_{-\frac{\pi}{\Delta w}}^{\frac{\pi}{\Delta w}}f(t).exp({-in\Delta wt)}dt\end{pmatrix} .exp(in\Delta wt)$$
then he took the limit as $\Delta w \longrightarrow 0$ which made the integral inside the braces become the Fourier transform formula
but how did the term $n \Delta w$ after we took the limit become just continuous $w$ ?
and the idea of Fourier series is to write the function in terms of orthogonal basis how our basis change after we took the limit , is it become a continuum of basis ? but that doesn't make sense because the basis must be n- dimensional